Find the inverse of the function [tex]y = x^2 - 12[/tex].

A. [tex]y = \pm \sqrt{x - 12}[/tex]

B. [tex]y = \pm \sqrt{x + 12}[/tex]

C. [tex]y = \pm \sqrt{x} - 12[/tex]

D. [tex]y = \pm \sqrt{x} + 12[/tex]



Answer :

To find the inverse of the function [tex]\( y = x^2 - 12 \)[/tex], we need to follow these steps:

1. Start with the original function:
[tex]\[ y = x^2 - 12 \][/tex]

2. Swap [tex]\( x \)[/tex] and [tex]\( y \)[/tex]. This step is due to the definition of an inverse function, which exchanges the roles of [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
[tex]\[ x = y^2 - 12 \][/tex]

3. Solve for [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex]. First, isolate [tex]\( y^2 \)[/tex] by adding 12 to both sides:
[tex]\[ x + 12 = y^2 \][/tex]

4. Take the square root of both sides. Remember that taking the square root introduces both a positive and a negative solution:
[tex]\[ y = \pm \sqrt{x + 12} \][/tex]

Thus, the inverse function is:
[tex]\[ y = \pm \sqrt{x + 12} \][/tex]

Among the given options, the correct inverse function is:
[tex]\[ y = \pm \sqrt{x + 12} \][/tex]